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Ineq in Geo again

Source: 1997 Korea National Olympiad #5

March 18, 2018
geometryinequalities

Problem Statement

Let a,b,ca,b,c be the side lengths of any triangle △ABC\triangle ABC opposite to A,BA,B and C,C, respectively. Let x,y,zx,y,z be the length of medians from A,BA,B and C,C, respectively. If TT is the area of △ABC\triangle ABC, prove that a2x+b2y+c2z≥3T\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}\ge\sqrt{\sqrt{3}T}
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